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Linear Algebra : Linear Algebra

Study concepts, example questions & explanations for Linear Algebra

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Example Questions

Example Question #11 : Range And Null Space Of A Matrix

What is the largest possible rank of a $7 \times 2$ matrix?

Possible Answers:

None of the other answers

Correct answer:

Explanation:

The rank is equal to the dimension of the row space and the column space (both spaces always have the same dimension). This matrix has seven rows and two columns, which means the largest possible number of vectors in a basis for the column space of a $7 \times 2$ matrix is , so this is the largest possible rank.

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Example Question #12 : Range And Null Space Of A Matrix

What is the smallest possible nullity of a $7 \times 2$ matrix?

Possible Answers:

None of the other answers

Correct answer:

Explanation:

According to the Rank + Nullity Theorem,

Rank(A)+Nullity(A)= columns

Since the matrix has columns, we can rearrange the equation to get

Nullity(A) = 2- Rank(A)

So to make the nullity as small as possible, we need to make the rank as large as possible.

The rank is equal to the dimension of the row space and the column space (both spaces always have the same dimension). This matrix has three rows and five columns, which means the largest possible number of vectors in a basis for the row space of a $7 \times 2$ matrix is , so this is the largest possible rank.

Hence the smallest possible nullity is 2-(2) =0 .

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Example Question #302 : Operations And Properties

A matrix with five rows and four columns has rank 3.

What is the nullity of ?

Possible Answers:

Correct answer:

Explanation:

The sum of the rank and the nullity of any matrix is always equal to to the number of columns in the matrix. Therefore, a matrix with four columns and rank 3, such as , must have as its nullity 4- 3= 1 .

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Example Question #14 : Range And Null Space Of A Matrix

$C(-\infty , \infty)$ , the set of all continuous functions defined on $(-\infty, \infty)$ , is a vector space under the usual rules of addition and scalar multiplication.

True or false: The set of all functions of the form

f(x) = 2nx + n ,

where is a real number, is a subspace of $C(-\infty , \infty)$ .

Possible Answers:

False

True

Correct answer:

True

Explanation:

A subset of a vector space is a subspace of that vector space if and only if it meets two criteria. Both will be given and tested, letting

$S = \left \{ f(x)| f(x) = 2n x + n , n \in \mathbb{R}\right \}$ .

This can be rewritten as

$S = \left \{ f(x)| f(x) = n( 2 x +1) , n \in \mathbb{R}\right \}$

One criterion for to be a subspace is closure under addition; that is:

If $a, b \in S$ , then $a+b \in S$ .

Let $f, g \in S$ as defined. Then for some real n, m :

f(x) = n( 2 x +1)

g(x) = m( 2 x +1)

(f+g)(x) = f(x) + g(x)

= n( 2 x +1) + m( 2 x +1)

= (n + m)( 2 x +1)

It follows that $f+g \in S$ .

The second criterion for to be a subspace is closure under scalar multiplication; that is:

If $a \in S , c \in \mathbb{R}$ , then $ca \in S$

Let $h \in S$ as defined. Then for some real :

h(x) =p (2 x +1)

(ch)(x) = c (h(x))

$= c \cdot p (2 x +1)$

=( c p )(2 x +1)

It follows that $ch \in S$

, as defined, is a subspace of $C(-\infty , \infty)$ .

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Example Question #12 : Range And Null Space Of A Matrix

$C(-\infty , \infty)$ , the set of all continuous functions defined on $(-\infty, \infty)$ , is a vector space under the usual rules of addition and scalar multiplication.

True or false: The set of all functions of the form

f(x) = 3nx + n + 1 ,

where is a real number, is a subspace of $C(-\infty , \infty)$ .

Possible Answers:

False

True

Correct answer:

False

Explanation:

A subset of a vector space can be proved to not be a subspace of the space by showing that the zero of the space is not in .

Let be the subset in question, and let f(x) = 0 , the zero function, which is in $C(-\infty, \infty)$ . This cannot be expressed as f(x) = 3nx + n + 1 for any . It if could then

3n = 0 , in which case n = 0 ,

and

n+ 1 = 0 , in which case n = -1 .

By contradiction, $0 \notin S$ . is not a subspace of $C(-\infty, \infty)$ .

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Example Question #13 : Range And Null Space Of A Matrix

$C(-\infty , \infty)$ , the set of all continuous functions defined on $(-\infty, \infty)$ , is a vector space under the usual rules of addition and scalar multiplication.

True or false: The set of all functions of the form

f(x)= |nx |

where is a real number, is a subspace of $C(-\infty , \infty)$ .

Possible Answers:

True

False

Correct answer:

False

Explanation:

Let $S = \left \{ f(x)| f(x) = |nx | , n \in \mathbb{R}\right \}$ .

We can show through counterexample that this is not a subspace of $C(-\infty , \infty)$ .

Let f(x) = |3x| . This is an element of .

One condition for to be a subspace of a vector space is closure under scalar multiplication. Multiply f(x) by scalar . The product is the function

g(x) = -1 f(x) = -|3x| .

$g(x) \ne S$ .

This violates a criterion for a subspace, so is not a subspace of $C(-\infty , \infty)$ .

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Example Question #12 : Range And Null Space Of A Matrix

$C(-\infty , \infty)$ , the set of all continuous functions defined on $(-\infty, \infty)$ , is a vector space under the usual rules of addition and scalar multiplication.

True or false: The set of all functions defined on $(-\infty, \infty)$ with inverses is a subspace of $C(-\infty , \infty)$ .

Possible Answers:

False

True

Correct answer:

False

Explanation:

Let be the set of all functions on $(-\infty, \infty)$ such that $f^{-1}$ is defined.

A sufficient condition for to not be a subspace of $C (-\infty, \infty)$ is that the zero of the set - which here is the zero function f(x) = 0 - is not in . $f \notin S$ because the zero function - a constant function - does not have an inverse ( f(0) = f(1) = 0 , violating a condition of an invertible function). It follows that is not a subspace of $C (-\infty, \infty)$ .

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Example Question #12 : Range And Null Space Of A Matrix

$C(-\infty , \infty)$ , the set of all continuous real-valued functions defined on $(-\infty, \infty)$ , is a vector space under the usual rules of addition and scalar multiplication.

Let be the set of all functions of the form

$f(x) = \left\{\begin{matrix} 0,& x< 0\\ nx,& x \ge 0 \end{matrix}\right.$

True or false: is a subspace of $C(-\infty , \infty)$ .

Possible Answers:

True

False

Correct answer:

True

Explanation:

A set is a subspace of a vector space if and only if two conditions hold, both of which are tested here.

The first condition is closure under addition - that is:

If $a , b \in S$ , then $a+ b \in S$

Let $f , g \in S$ as defined. Then for some $n, m \in \mathbb{R}$ ,

$f(x) = \left\{\begin{matrix} 0,& x< 0\\ nx,& x \ge 0 \end{matrix}\right.$

and

$g(x) = \left\{\begin{matrix} 0,& x< 0\\mx,& x \ge 0 \end{matrix}\right.$ .

$(f+ g )(x) = \left\{\begin{matrix} 0+ 0,& x< 0\\nx + mx,& x \ge 0 \end{matrix}\right.$

$(f+ g )(x) = \left\{\begin{matrix} 0 ,& x< 0\\(n + m)x,& x \ge 0 \end{matrix}\right.$ ,

$f +g \in S$ . The first condition is met.

The second condition is closure under scalar multiplication - that is:

If $a \in S$ and is a scalar, then $ca \in S$

Let $h \in S$ as defined. Then for some $p \in \mathbb{R}$ ,

$h(x) = \left\{\begin{matrix} 0,& x< 0\\ px,& x \ge 0 \end{matrix}\right.$

For any scalar ,

$c \cdot h(x) = \left\{\begin{matrix} c \cdot 0,& x< 0\\ c \cdot px,& x \ge 0 \end{matrix}\right.$ ,

and

$(c h)(x) = \left\{\begin{matrix} 0,& x< 0\\ (c p)x,& x \ge 0 \end{matrix}\right.$

$c h \in S$ . The second condition is met.

, as defined, is a subspace.

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Example Question #19 : Range And Null Space Of A Matrix

$C(-\infty , \infty)$ , the set of all continuous real-valued functions defined on $(-\infty, \infty)$ , is a vector space under the usual rules of addition and scalar multiplication.

Let be the set of all functions of the form

$f(x) = \left\{\begin{matrix} 2nx+2n , & x \le - 1 \\ 0, & -1 < x < 1\\ -nx+n & x \ge 1 \end{matrix}\right.$

for some real

True or false: is a subspace of $C(-\infty , \infty)$ .

Possible Answers:

True

False

Correct answer:

True

Explanation:

A set is a subspace of a vector space if and only if two conditions hold, both of which are tested here.

The first condition is closure under addition - that is:

If $a , b \in S$ , then $a+ b \in S$

Let $f , g \in S$ as defined. Then for some $n, m \in \mathbb{R}$ ,

$f(x) = \left\{\begin{matrix} 2nx+2n , & x \le - 1 \\ 0, & -1 < x < 1\\ -nx+n & x \ge 1 \end{matrix}\right.$

and

$g(x) = \left\{\begin{matrix} 2mx+2m, & x \le - 1 \\ 0, & -1 < x < 1\\ -mx+m & x \ge 1 \end{matrix}\right.$

Then

$(f+g)(x) = \left\{\begin{matrix}( 2nx+2n)+ ( 2mx+2m) , & x \le - 1 \\ 0+0 , & -1 < x < 1\\ (-nx+n) + (-mx+m) & x \ge 1 \end{matrix}\right.$

$(f+g)(x) = \left\{\begin{matrix}( 2nx+2mx )+ ( 2n +2m) , & x \le - 1 \\ 0+0 , & -1 < x < 1\\ (-nx+(-mx)) + (n+m) & x \ge 1 \end{matrix}\right.$

$(f+g)(x) = \left\{\begin{matrix} 2(n + m)x+ 2 ( n +m) , & x \le - 1 \\ 0 , & -1 < x < 1\\ -(n +m)x + (n+m) & x \ge 1 \end{matrix}\right.$

$f +g \in S$ . The first condition is met.

The second condition is closure under scalar multiplication - that is:

If $a \in S$ and is a scalar, then $ca \in S$

Let $h \in S$ as defined. Then for some $p \in \mathbb{R}$ ,

$h(x) = \left\{\begin{matrix} 2px+2p , & x \le - 1 \\ 0, & -1 < x < 1\\ -px+p & x \ge 1 \end{matrix}\right.$

For any scalar ,

$c h(x) = \left\{\begin{matrix} c (2px+2p) , & x \le - 1 \\ c \cdot 0, & -1 < x < 1\\ c(-px+p) & x \ge 1 \end{matrix}\right.$

$(c h)(x) = \left\{\begin{matrix} 2(cp)x+2 (c p) , & x \le - 1 \\ 0, & -1 < x < 1\\ -(cp)x+cp & x \ge 1 \end{matrix}\right.$

$c h \in S$ . The second condition is met.

, as defined, is a subspace.

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Example Question #12 : Range And Null Space Of A Matrix

If is an $m \times n$ matrix, find

dim(row(A))+dim(col(A))+dim(null(A))+dim(null(A^T)).

Possible Answers:

2n+2m

n+m

|n-m|

Correct answer:

n+m

Explanation:

Since a basis for the row space and the column space of a matrix have the same, number of vectors then their dimensions are the same, say .

By the rank-nullity theorem, we have rank(A) +Null(A) = n , or same to say

dim(row(A))+dim(null(A)) = n .

r +dim(null(A)) =n .

Hence dim(null(A))=n-r .

Finally, applying the rank-nullity theorem to the transpose of , we have

rank(A^T) +Null(A^T) = m , or the same to say

dim(row(A^T)) +dim(null(A^T)) = m .

r + dim(null(A^T))= m (The row space dimension of is the same as its transpose.)

dim(null(A))=r-m .

Adding all four of our findings together gives us

r+r+n-r+m-r = n+m .

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Linear Algebra : Linear Algebra

Study concepts, example questions & explanations for Linear Algebra

All Linear Algebra Resources

Example Questions

Example Question #11 : Range And Null Space Of A Matrix

Example Question #12 : Range And Null Space Of A Matrix

Example Question #302 : Operations And Properties

Example Question #14 : Range And Null Space Of A Matrix

Example Question #12 : Range And Null Space Of A Matrix

Example Question #13 : Range And Null Space Of A Matrix

Example Question #12 : Range And Null Space Of A Matrix

Example Question #12 : Range And Null Space Of A Matrix

Example Question #19 : Range And Null Space Of A Matrix

Example Question #12 : Range And Null Space Of A Matrix

All Linear Algebra Resources

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